hi there today we're going to do section 8 f in class activity 8f and uh we're looking at the connection between the binomial and normal distribution so we're looking at that a little more closely and very specifically we're going to look at n and and the sample size so let's get started hope you guys are doing well today share my screen all right so um in a previous class activity we learned about the role of p in the binomial distribution so that that lowercase p is going to be reserved for um the probability of a single success so this is all for binomial i'm just going to it helps me to write this down maybe you don't need this known meal distributions if you hear that panting in the background it's my dog not me binomial distributions p equals the probability of success of a single attempt so um they're independent events so they're we're going to be treating basketball as if it's an independent event we know it's not true but we're we're looking at models so we're going to treat the basketball star as if he's kind of like a robot and what's the probability that he in any one attempt at getting a free throw he's going to make it that would be the probability of success and we're going to be using um decimals as we usually do this will always be a decimal so i'll just put that always a decimal okay but today so that was in a previous today we're going to look at n which is it is sample size but in this case it's going to be the number of attempts so you know the probability of him making a single basket what's the probability that out of seven he makes six baskets things like that that's what the probability and distribution is really wonderful for um sorry the binomial distribution is really wonderful and n is what we're focusing on today but as a little review in the preview activity you looked at the role of p in the distribution so i'm going to go and i'm going to look at the distribution here i'm going to open it up so this right here is a lovely picture of when p is equal to 0.5 and you can read that in the title there p said it point five and if you have p being close to fifty percent you get this nice bell-shaped curve uh it might be a little chunky but you always get that nice symmetrical distribution so um we played around with that so what we noticed in the previous preview is that p close to 0.5 is symmetrical p far away from 0.5 50 percent is skewed more skewed so let's just let's just revisit that for a second i'm going to open up my binomial distribution which of course i should have bookmarked and uh p is set at 0.5 to begin with you'll see that right there and this n right now is 6. and if we scroll up we can see oh look at that a beautiful um a beautiful symmetric distribution so let's change p to something far away from far away from 50 so how about 80 because we're so used to um having distributions that are skewed right and i don't know why i do that but so i'll set it at 80 and sure enough look what happened to our distribution it's skewed to the left and that actually makes sense because if the probability of of success is high 80 percent then it's more likely if you're looking down here if your eyeballs are right there it's more likely that you will get more successes than not so the probability of having no success is of zero is pretty small and that's why that first column is really really low so we just checked this out that yes it's really true that if p is far away from 50 um you're going to end up with a more skewed distribution but the question here is recall what you learned what did you discover about the role of n in the shape of the distribution of the binomial distribution so here we have we scroll up here i don't want to be able to see i can't can i shrink this no okay so i'm going to just scoot it up so you can barely see n and it's right in the tippy top corner of the um i'll just scoot it down a little bit so that you can see there's n c n right there i'm gonna just slip it up there so that was n equal to five or six what happens if you make n bigger and i'm just going to slide in still it's still skewed at 42 it's a little tiny bit skewed to the left then i'm looking here my green eyes looks like it's more oh more skewed the smaller it gets the more skewed it is and the bigger it gets the more symmetric it becomes and the more bell-shaped it becomes so oftentimes these questions are your own opinion but in this case it's not your own opinion i want you to write this down if you haven't already as n gets bigger the shape of the distribution gets more symmetrical and in fact we can go one step further that doesn't just symmetrical could be rectangular it gets more bell-shaped or normal so if you have a p that is far away from point five you can still get a nice bell-shaped binomial distribution you just have to have a bigger um sample size it has to be bigger and bigger and now sample size is really trials number of attempts at a situation okay so and that is exactly what is stated here for n large the binomial distribution can be approximated to the normal distribution now the problem here is enlarge is a very why didn't they just say a particular size for n because it depends on what your p is if your p the closer your p is to 50 the smaller your sample size can be to achieve that bell shape but we'll get to a rule of thumb really soon but you're just noticing the pattern that if i make my sample size smaller [Music] my distribution loses that bell shape if i make my sample size bigger my distribution becomes more bell-shaped and if i if i usually you can't play around with p but if i had 50 it doesn't matter how big my sample size is i'll still get that nice bell shape but it's chunky it's jaggedy it's not smooth so bigger sample size is better except it's also more expensive so we want to know what is big enough is what we're going to be looking at so the other thing that you want to notice is that the center of your distribution the mean of your distribution for a binomial distribution is going to be can be used to describe the most likely of values and if you have a 50 success rate which is right here then isn't it most likely that you'll get 50 of your baskets in and sure enough that's what you'll see down at the bottom of this binomial distribution and you can check it out the um the formula for the center mu mu mu equals n times p um and p whatever the true population is so the center the average the population average uh and that makes sense you know it's it's percent times amount equals the amount of successes so if you if you have a 50 success rate and you do a hundred attempts it's likely you'll get fifty if you do five at four attempts it's likely you'll get two so this is the most likely value and you can see it by seeing the peak in your distribution so that is the uh formula for the mean which i think is easily understood the formula for the uh standard deviation so it's population standard deviation so it's going to be sigma and it is equal to the square root of the sample size times the probability of success times the probability of failure and i'm not really going to explain that one i think it has been touched on and some um some previews but we're just going to leave that is the formula so if i ask you for the standard deviation of a binomial distribution um you can go here and find that formula okay the role of the continuity correction so this is the other thing that we're going to be talking about today is this thing called the continuity correction i just want you to be exposed to what it is and why we do it i i am never going to ask you to apply the continuity correction in an exam because honestly as a statistician i was never asked to use the continuity correction my husband's a data scientist and he was never asked to do it so i think it's maybe a theoretical thing that i want to show you in case you come to a higher level class and they mention it i don't want you to be intimidated because it's really just a little tiny thing but we won't be using it on exams or quizzes okay by the time this section is over you'll be able to determine when the normal distribution can be used when the normal distribution can be used to approximate the binomial distribution why would we want to do that because the normal distribution is so much easier to work with and because so much data follows a normal distribution so a lot of statisticians just that's all they apply all day long so and there's a really really short answer to this when can this happen it's really simple when n is big enough so if you have a big enough sample size um you can be sure that your binomial distribution can be approximated approximated using the normal distribution so but you're like well god big enough that doesn't sound very scientific that sounds very very vague and i'm gonna the rule of thumb is right down here n p has to be greater than 10 and n one minus p has to be greater than ten what this really breaks down into is np is the number of successes and remember success is a very subjective term if you're studying rare diseases that kill people and a success is you have someone with that disease so the number of successes and this right here and one minus p is the number of failures and they both have to be bigger than or equal to 10 half has to be number has to be at least there's that pesky at least at least means that number or higher 10. and here also has to be at least 10. okay so so when you multiply your sample size by your proportion you got to get at least 10 which means if you're studying baskets he's he the player has to get at least 10 baskets in the whole game and the whole season whatever you're focusing on in terms of disease a really really rare disease if you're studying a really really rare disease like um i was just looking up uh one of those if you google the one where people age prematurely so you see those little five-year-olds and they look like they're 100 years old it's a very rare disease and it's hard to study because it's so rare and you have to get when you get a pool of people you have to be sure that at least 10 of them will have this disease so if it's one in a million you've got to study 10 million people to get to get any kind of valid results so that's hard and then also but also the number of failures has to be um greater than i don't leave this eyeball anymore greater than or equal to 10. so an example for this one is um believe it or not herpes is really hard to study because not if you look at people over the age of 35 i now this was about 10 years ago maybe it's gotten better maybe it's gotten worse but they adults sexually active adults they were thinking over the age of 35 uh 88 have a form of herpes so think about that when an older person approaches you it says hey baby that 88 have a form of herpes so that means it's to find if you get a pool of people failure if you're studying herpes a failure is they don't have herpes you have to have enough people in your pool that you're studying so that at least 10 don't have illness so you've got to have that kind of sweet spot of enough habit and enough don't to have valid results so that is the rule of thumb and i'm just going to highlight it and if i ask you this is what big enough means right here big enough means n p is greater than or equal to 10 and n one minus p is greater than or equal to ten and we're going to be um studying that we're going to be practicing that really soon actually so um so that's that so let's go we're now talking about i think we're talking about this example right here we've got a superstar basketball player who is really good at free throws um recall the free throw scenario in preview 8f suppose um suppose that over the course of a season paul george who's a top thrower um free throw per shooter uh will get 300 free throws okay so he's got our 300 free throws so that means he's going to somehow get people to foul him so that he gets to shoot at the basket without people charging him 300 times that's pretty successful right there um we know that the top shooters in the league have a probability of about 90 percent okay so we've got 90 percent here um can the binomial distribution be approximated by the normal distribution so if you're talking about the binomial distribution you need to identify what p is and you need to identify what n is because you can't do this unless you can't have a binomial distribution until you recognize that so n is the number of attempts the number of trials the sample size and p is the probability of success of a single attempt and i hope if you re-read it you're going to see that the p is this 90 so this is an amazing shooter he has a probability of a success of a single attempt just 90 of the time and he gets to shoot 300 times so the question is um free throw can the binomial distribution be approximated by the normal distribution that is code for is n big enough and what you have to do is you have to check the sample size but you don't just have to check the sample size you have to check check the sample size you have to do it twice both conditions have to be fulfilled what are the conditions the conditions are um and p has to be greater than or equal to 10 and and it's not an or both has to happen and 1 minus p has to be greater than or equal to 10. and i am right now going to put a little question mark above here these are question marks because we're checking to see if that's true so i'm just going to go ahead and plug in my values plug in my values so n is 300. that's a little hard to read n is 300 so i'm popping that in there and p oh that's gonna ruin my work p is 0.9 okay and 300 times 0.9 is going to be i should have this in my head 270. so that's going to end up being 270. and is 270 greater than or equal to 10 yes so you're happy but you're not done because that what that tells you is that if he's allowed to to attempt 300 times we would expect him to get it about 270 times which is well above the threshold of 10. so we're we're happy but we're not done now we've got to do it with the complement so n is still 300. i apologize for that being so small it's too small let's give it a little room here um 300 and one minus p i like to work that out in my head if 90 is successful then that means that uh 10 is a failure but if you don't want to do that i'll do 1 minus 0.9 and work it out that ends up being point 10 or 10 which is the number of fails and when you multiply that by 300 it's going to be and while that is a smaller number it is greater than or equal to 10. so yay so you got both checks and since you have both checks are correct you can say yes um yes the binomial i don't really have a lot of room here yes use normal because n is big enough smiley face and you're happy because the calculations for normal distribution are so much easier all right so now um we're going on to talk about this continuity correction and um what's useful for the continuity correction is let's think about uh his distribution so i'm going to put his actual distribution in he has a 90 success rate i'll move that to 90 which makes it very skewed but we just established that his oh shoot i can't do that sample size there i'm going to go to probabilities because that'll give me a little more freedom uh but to do the probabilities i have to know what the center of my distribution is and i have to know what um so the center of my distribution no i don't that's a different distribution so number bernoulli trials how many times is he going to shoot that basket 300 times yay and um 90 success and let's look oh that looks beautiful that completely verifies what we just said right here and big enough you get that nice l-shaped curve but if i ask you to show me you have to do the calculations you can't just show me a pretty picture you have to do the calculations so if i'm interested in him getting exactly um exactly 700 270 baskets in a season where he attempted 300 times that probability i'm going to go up here and i'll put in 270 it does it for me okay so it's do you see that column right there [Music] okay you see that column right there and i'm going to do a little sketch of that so probability of exactly 270 baskets what i want to visualize is i want to visualize just a chunky distribution and i'm not going to draw all of these to get the idea and the most common value is 270 that's reflected in the peak i'll put a 270 here and we calculate that area well we use the binomial distribution and they got if you look up there 7.66 percent so the chances of it being exactly 270 is actually pretty small even though it's the most common the most likely value and if on an exam or on a quiz or on a test i ask you what's the probability i actually do want you just to write the percentage that's under the title because it's it's so much easier uh it's less work for you so that's going to be p of x equaling 270. okay and that's the binomial that is the exact um that is the exact so it's not approximated this is exact because we're using the binomial distribution but if you want to use the normal distribution instead because we said it was big enough right so if you want to use the normal distribution instead you're going to get this nice spell shape i'm trying to make it nice and the center is going to be that same 270 and the spread you can get from that pretty we didn't really explain the formula but the spread is right here and if you put in the values sigma is equal to n is 300 p is 0.9 1 minus p is 0.1 and when you run that through the calculator and i do want you to go to three places past you do it and i'll do it i got um sigma equals 5.196 okay so not very intuitive but what that's telling you is 5.1 nine six that's telling you about five so let's let's not be so precise it's about five so that's saying that we expect this superstar to get 68 of the time he's gonna get between 265 and 275. on a season but the question is how much is he going to get what's the probability of him getting exactly 270 if you're dealing with the normal distribution so exactly 270 is going to be all of these little dots right here but if you're doing a continuous distribution you're assuming that there's a there's a hundred million attempts um or a hundred million data points and so uh the way you calculate probability for a normal distribution is it's the area under the curve and so here x equals um probability well so we're talking about individual shots i'm not going to do it's too fussy but if i want to do that i'm going to have to to figure out that the area of that red line so we know probability is area for this well let's move to let's move away from the binomial distribution and let's use the normal distribution so i'll go to the minimum distribution here we go and it seems like everything is checking along just fine and i want to actually find a probability and the mean we just figured out is 270 good and the standard deviation we just figured out using that horrible formula the standard deviation is right here it's 5.196 i hope it reads that is that i think it's using a decimal a comma there so i'm going to try again 5.196 i hope that's right we'll know in a minute because if it doesn't answer my question i'll know it um my eyes are really bad so we want and if you look here it gives you is x bigger than 270 is x smaller than 270 is x in between two values it doesn't give you the option of equal to 270 but i'm gonna i'm gonna jerry-rig it so i'm gonna go here oh and it's not it is not letting me do that because it's not reading this right let's try again 5.196 and let's see if it does this okay good okay so that's just a glitch in my program so i'm gonna let both a and b be 270 because i want to get the probability of exactly 270 and it didn't give me that so i'll just cheat by putting a 270 in here and a 270 in here and it says look at that it gives me that the probability is zero do you see that if you look right here you can see under the title it says the probability of x being between 270 and 270. so exactly 270 is zero and that's exactly what's problematic about for when you're thinking about a normal distribution you cannot think of one particular value and we've talked about that before where women's heights the average woman is five foot four but no woman in the world is exactly five foot four so you've got to create a range so you want to create the equivalent range of what you see here and here that width is about one so what you want to do instead is you want to go a little bit on both ends so i'm going to go a tiny bit on this side and i'm going to go a tiny bit on this side and by a tiny bit i'm going to say this and i'm going to go 0.5 this way 2 6 9.5 and i'm going to do the same thing on this side to kind of mimic i know that there is a probability so i'll do 2 7.5 on this side so i'll fix that we'll change this to 2 269.5 and we'll change this to 2 7 0.5 and it creates that now we can actually have an area associated with that and the probability of that according to the normal distribution is i don't know if you can see that oh well it's in the title um 7.67 is what they rounded it to be 7.67 well gosh here we got this and here we got this they're really close so that's a continuity correction so if you want a specific value you've got to create a little wiggle room on either side of it you cannot calculate the area if it is exactly a value you have to create an interval so go 0.5 in either way presto bingo and you're good to go so i am not going to do well how should i do this or not i don't think so i think we're gonna we're gonna skip this um because i i think this is good enough right here because the truth is that a good statistician is never going to try to find the probability of a single event when it's a continuous variable it's always going to be an interval and it's probably going to be more of a robust interval than one unit um then we would just use the binomial distribution so um we're not going to do examples of this i just want you to have um to have been exposed to continuity correction because the best thing to do is just know that the normal distribution is an approximation and so you you keep that in the back of your head that if you really want to be precise you can use the the binomial but i mean gosh the probability of success the exact probability is 77.66 and the estimated is 7.67 so i would say this is a pretty good option for um for practical reasons it's way better to use the normal distribution than the binomial okay so let's do a little bit of practice um now let's use the normal approximation with continuity correction um using uh so we're doing what we're doing right here okay we're using the normal distribution okay so i i'm doing that okay um now the steps that i recommend when i'm asking you to calculate probabilities the steps that are going to keep you safe steps one visualize the distribution two plot the observation and shade appropriately um three i just did i want you to and then use the technology then then use technology um and area equals probability okay so um for part a oh i should have written that up there oh well never mind uh for part a find the probability that george makes at least um 268 free throws in a season so you can't answer this question until you have an approximation and we're using the normal distribution so i'm going to go ahead and start by drawing it's kind of like this is what i'm going to hang all my information on because otherwise it gets a little overwhelming and so what we're plotting is how many baskets he's it's x equals number of successful baskets out of 300. so we know what we know and we don't want to abandon that information p equals 90 and we also know that n equals 300. so um can we be sure that it follows this shape we already in problem number um and problem number two we checked that the sample size was big enough so gay we're gonna we're good with that um so what is mu and what is sigma okay so we know that mu in general is np or pn it doesn't matter um so we know it's n times k and n is 300 and p is 0.9 so that ends up being 270. so our expected value is 270. that's how many baskets we expect him to get um for any given season and then i think i'll do sigma over here and i did do it but i think i'll just do it again sigma is this formula n p 1 minus p so it's going to be 300 times 1 minus p is 10 and then you square root it all and when you're done that's 5.196 okay so that right here is 5.1 okay so i'm just visualizing this you did you don't have to do that it's saying that we the give or take is about oh did i get that right 300 times 0.9 times 0.1 equals 27 and then i'm going to square root it yep that's right okay so there's my and so it gives me a sense if i'm gonna use the empirical rule i'm not gonna do the decimals i'm gonna just do two six five and two seven five that's my hump and then if i go out another five two i'm rounding a little bit um two eighty that give or take five means that we would expect that about 95 of all of his shots in any given season fluctuate between 260 and 280 it's give give or take a little bit of change all right uh find the probability that george will make at least um a hundred and 268 free throws so i v i visualize my distribution you didn't have to be as precise as i was just you know it looks like this you know the 270s in the center i'm now going to plot my observations so 268 is maybe about right here but we're five in so so it's just a guess 260 about right there 268 and we're talking about at least at least 268 so what does that look like to at least 268 is 268 269 270 oh i should be shading this way so i already know before i go to technology that um so i shade it appropriately and this if anything goes wrong this is probably what goes wrong so you've got to really unpack what at least at most all that is now use technology to find the area area equals probability so the probability here um is going to be the total is 100 that's going to be the fraction of that so let's uh let's go to so and for fun we could do the notation p x is greater than or equal to um 268. okay and so i've got it all here all i need to do is put in the 268 pick my right tail so if you have trouble with these symbols just go ahead and pick and say i'll pick this one and go okay that's not the shade that i want i'll pick this one that's the direction i want but of course the cut point which is the observation is not right i'll change that 268 does my picture that i drew here look like my picture over here yeah so that makes me happy and it says it's 64.98 percent yay um so we're not using the continuity correction did i say to do that next important use the with the continuity correction so i am crossing this out we're just because no one ever asks me to do that so we're just doing a straight up sketch approximation um so next let's use the normal distribution to approximate okay so here's my answer almost 65 chance okay so part b that was a b find the probability that george will make between those two values inclusive so i'm going to do the same thing rough sketch and you could this could just be in your head if you want so we know two seventies here and one of the observations is 256. so it is good to have a little rule of thumb here so that's 265 and that's 260. so 256. oh that's 255. so you had to go really far out in this direction it's about right there that's that one so i'm going to want you to pause and i want you to to plot this one on your distribution i want you to shade appropriately and then i want you to use the normal distribution to approximate what you think it would be keeping in mind that the true distribution is binomial but we're not fussing with that because we can blur it all together and it works well enough so pause and finish this and then check back with me okay so the second observation um is 284 so just rule of thumb because i said this is 275 this is 280 because i've rounded i've said that the spread is more is more like five why not because it's all approximation so this is 285 so 284 is to be right here about so it looks to me if i use the empirical rule which you don't want to use unless i expressly say to i'm going out not one not two almost three standard deviations in that direction and in the other direction i'm going out not one not two almost three standard deviations so i would expect it's 68 for one standard deviation 95 99 point something or another i can never remember that one but the probability of him getting both of those between that range it's almost a certainty it's almost a hundred percent i already know that before i use the technology so let's go ahead and use the technology we'll do the two tails and we've got um 256 and we have 284 and lo and behold it the picture checks out and the probability checks out so the probability that x is between those two boundaries um 255 56 and 284 is about and it is an approximation 99.29 of the time so in a sense you would say we would expect that he would get a value in that range 99.29 percent of the time and so we're not doing the continuity correction the continuity correction would ask you to pull it back just a little bit this way half a half a unit because you can't get exactly 55 so you would have 55 254.5 and pull it back a little bit this way and that would be a tiny bit more precise but i say who cares um it's good enough at this is good enough for me don't use the continuity correction because i never did ever all right maybe i didn't build a bridge or something where it had to be so precise but if you go on and you're building bridges you'll probably have to have another semester of uh statistics anyway i hope so anyway all right um so that was this answer okay find the probability that um george will make more than 285 free throws so this go ahead don't use the continuity correction follow these steps do this first and that way because if you notice my exams are not super time intensive so you draw the sketch then you use the technology and if your sketch is different than the technology or vice versa you can catch your error so go ahead and practice c um and practice d do c and d and um we're not doing continuity correction we're just gonna do straight up um okay so for c i'm going to draw the picture above and i'll do the probability below i actually i should probably do the picture below so find the probability that he'll make more than 285 so you really here's my sketch i know it's 270 and you know i don't exactly know where 285 is maybe i'm rushed so i'll just say 285 i don't know how many standard deviations away probability that he will make more so we don't care about exactly 285 whether it's more than or more than or equal to it's the same value because that orange line has no area so it has no probability so the area is going to go ahead and we're interested in this right here um and so let's go and throw it in we'll change that to i'm i'm going to just play around and see if i get yep that's the the one i want and it's 285 and be careful because you might put the 85 in first the 285 and it'll change it for you so that's it's saying hey that's really far out so the probability of x being greater than 285 is going to be um 0.19 percent so that's not 19 percent that's less than um that's almost that's really small it's less than one percent way less than one percent and by the way the probability that x is greater than or equal to 285 equals the same because this this line right here has no probability associated with it because the probability is area lines don't have occupy area so here's your answer right here um and if you wanted to convert that to a decimal it would be point zero zero one one so it's a really small area it's not twenty percent don't think that okay um so for part c what did they ask um would it be surprising or unusual for george if george were only to make um 230 free throws in a season so that is an exact answer only 234 free throws so i would say if you wanted to answer this question you could do the binomial but you could also think about so that would be a valid thing is to to switch it to binomial because they were asking for a precise answer um so where is the binomial there it is and here we could throw that in um and you could work it out but oh that's such a pain it's such a pain to do that right so the other thing you could do is you could ask yourself how far away is this from the center here's the center right how far away is 230 what's a measure that tracks how far something is from the center if you're dealing with the bell shape which we know we are you can use z-score so go ahead and for this one so if i ever ask you how unusual is a certain observation a really good approach if as long as you know that you're dealing with something that can be approximated with a normal distribution you can go ahead and do the z-score on it to see how far how many standard deviations it is from the center so um so go ahead and pause and work that out so z of 230 for this superstar chris what's his name chris is his name chris brown i should know his name he's a superstar but i don't i'll just i'm we're on first name basis me and chris so z230 is going to be observation minus center over spread so for my distribution i know that the center is 2 270 and i know that the spread this distance right here was 5.196 so i was being sloppy by doing 5 but that's okay so that's 5.196 and the actual observation is 230. so if he's really you know really consistent and his success rate sticks at 90 percent what's the probability that in one season he'll have this happen so if we work this out i haven't worked that out let's do it and is it appropriate to be calculating z-scores i think so because i know that my distribution is very close to being normal so this is going to be negative 40 over 5.196 divided by well that's not looking very good 1 6. so it is so it's an approximation let's put little wiggly sounds of science here signs here negative eight seven point six nine eight so if we are going to do i did this an entirely different way in my notes would it be surprising or unusual that george were to only make two hundred there's one of those explain i did a whole different way here i like this better though um i know that i can visualize now that if this is the distribution he's not one standard deviation below he's not two standard deviations below he's not three standard deviations below four five i've run out of room his his horrible um 230 isn't even on the screen like i'd have to extend my screen so 230 is more than um seven almost eight let's say is almost is almost eight standard deviations below if we assume that this can be approximated by a normal distribution so um i'll throw him approximately so yes this would be a very unusual result so i just i chose z-scores um surprising my go-to is z-scores um you could use the empirical rule and go well if i go three standard deviations out it's um three standard deviations is going to be 5 10 15 below so that's going to be 2 55 and this is even lower than that we know that most data is within three standard deviations this is way further away than that so do me a favor and some of you did this on the exam um highlight your punch line for me so that i'm not uh desperately looking um for for to give you points the easier you are on me to find your solutions the nicer i am on you when you need partial credit okay how does the answer in part a for the exact binomial probability you derived in question 2e okay i do not know what they're talking about there so how does the answer in part a question preview oh question two part e preview assignment so we're interested in the probability that x is greater than or equal to 268. so let's backtrack because i i don't want to have to so the question here if we look at those two questions is what's the probability that x is greater than or equal to 268. we're not going to use the continuity correction we're just going to just we already go oh n equals 300 is big enough so the big news of what to learn is that um so using first we'll use a binomial by non-meal uh calculation we know this is a binomial distribution because we've got a concrete number of successes uh probability of success we've got a concrete number of trials and we're treating this as if it's independent events meaning if he misses one basket it's not going to mess up in his head for the next one so we've got our binomial distribution here so we know that n equals 300 we know that p equals 0.9 okay so n equals hundred and p is point nine good and so there will looks beautiful normal but we're using binomial and we want a value of 268. [Music] and uh we don't want equal we want greater than okay that's not right so we'll pick the next one now i want to just be careful for a minute here because i think the question was greater than or equal to or is it greater than ah i'm gonna have to actually go check with uh the preview assignment so i'm gonna pause this for a second and i'll be right back recording in progress well my computer is having a nervous breakdown and so i'm going to wrap this recording up before it crashes i wasn't able to check this but i do not i think this is a typo so let's just cross this out um i know it wasn't that's it's not question two so let's just um we have um calculated the probability well let's just do it now so um we're going to the calcul it's going to be exactly we're including 268. so when i look at my picture over here it's not greater than 268 it's greater than or equal to 268. so 268 is included in the shaded and we can see it's the orange area the probability is going to equal so p x is greater than or equal to 268 and this is not a binomial this is not an approximation it equals 69.17 so it's very it's more it's more likely than not that he's going to get at least 260 fastest 68 baskets so that's if we do the binomial exact now did we do this already for look we did it already up here for the um for the approximation we got 260 but we got 64.98 um so using the normal distribution approximation with no continuity correction we got p x is greater than or equal to 260 260 8 is approximately equal to and i'm coming up here 268 64.98 how do those compare um well they're not perfect actually i think maybe we should have done the continuity correction um if you want to do the continuity correction so not perfect i hope i didn't do anything wrong 64.98 that's right [Music] yeah so let's do the continuity correction you'll never have to do this though in my class so using the continuity correction okay so um we've got this picture and i think i'll just transport it here okay copy just gonna bring it down here paste image okay make it a little bigger so there it is um and what we're saying is that two since 268 doesn't have any area associated with it we're gonna make it a little bit bigger because that area associated with that is zero so we'll just pull it back a tiny bit and we'll make it 267.5 i find it hard to believe that that's going to be that big a difference but so because there's zero area associated with 268 we don't want to lose that so we'll just make that a little bit bigger and so now we'll do um what's the probability so now we'll count it'll be a tiny bit bigger so p x is greater than or equal to 267.5 to give it that little wiggle to capture the 268 and now i will go to the normal distribution because we're approximating using the continuity fraction and we've had a mean of 270 that's what's expected and the standard deviation of 5.169 it was 196. oh that wasn't my error nope doesn't look like it and we want an upper tail grab the other tail and we're going to let x now to allow for we lost a little bit of area 267.5 i still don't have the tail the right way this is greater than i'm having problems with my my computer is not my internet's not working i hope this is recording why won't it switch well um i don't know what to tell you pause it again hope that my internet comes back okay well this this is going to give you will give you a little more area and will be better but not required in this class and right now the only thing my computer can do is record so i can't seem to access i cannot seem to change anything here so it's frozen so not kind of restart it because i don't want to lose this recording so we're done uh i'm just going to say this is pretty good enough this compared to this whether you do normal approximation with wiggly lines or you do the precise with the binomial and we're going to be moving away from binomial and we'll get bigger and bigger sample sizes so that it gets more and more approximately normal so let's review what's important here and i'm sorry it didn't end well but these things happen if n is large enough the binomial distribution can be approximated using the normal distribution and we love the normal distribution because it's easier to use when your computer is working which mine wasn't right then um so if you're dealing with a binomial um the mean is going to be np and the standard deviation is going to be this formula right here um so a little bit obnoxious i didn't really explain it but um but we're gonna we'll get a better explanation pretty soon but n has to be big enough and the rule of thumb is right here and p is greater than or equal to 10 and n one minus p is greater than or equal to ten so you have to do check check uh to do this oh my notation is slowing down too i think my computer has a virus so um so no the most important thing to know is n big enough okay and um you were introduced to something called a continuity correction it's where you add a little bit of area to acknowledge that the area of a single observation is equal to zero but you know that single observation actually has a probability associated with it so you give it a little bit more by either taking away or adding 0.5 units to the discrete variable and but we are going to do that in this class and so um so that's it okay i hope this recording worked out we'll see you